Tardyons and Tachyons in Direct-action Electromagnetism Michael Ibison Institute for Advanced Studies at Austin 4030 West Braker Lane, Suite 300 Austin, TX 78759, USA [email protected]Tel: 512 346 1628 Fax: 512 346 3017 Abstract We make a case for the extension of classical electrodynamics to the domain of superluminal and time- reversing motion of charges. A key ingredient is the absence of mechanical mass-action () =− − ∫ 2 1 mech I m dt v t which usually prohibits superluminal motion. Using the direct-action version of EM we argue for an action wherein all inertia is derived from electromagnetic interactions. A novel action is proposed in which self-action is present for (superluminal) tachyons but absent for (sub-luminal) tardyons. Consequently, in this model, electrons must acquire their mass from interaction with other charges, whilst superluminal charges acquire their mass from self-action. There is some discussion of a possible correspondence between tachyons and quarks. 1
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Tardyons and Tachyons in Direct-action Electromagnetism
interact remains satisfied. With (12), tardyons, because they do not self-interact, have zero electromagnetic
mass. One could, of course, now restore the mechanical mass to the action in keeping with one of the original
motivations for exploring the direct-action version of EM. But this would be rather ugly; it would introduce a
mass scale for tardyons but none for tachyons (and perhaps, therefore, baryons). Eschewing that possibility,
without an additional mechanical mass, the tardyon is completely massless. (This possibility was dismissed in
[9].) In summary, this approach has the multiple benefits that:
0∆ →
i) The bare mass may possibly be dispensed with entirely;
ii) The diagonal term in the double-sum can be retained without incurring self-action and therefore
without infinite self energy;
iii) Self action is present in the (possibly intermediate) pair creation processes identified by Feynman as
crucial for the correctness of QED scattering calculations [10].
A massless electric charge with position ( )tr and velocity ( ) ( )t ≡v r t must satisfy the Lorentz force law
[11,12]
( )( ) ( ) ( )( ),t t t t t,+ ×E r v B r 0= . (14)
Except at time-reversals, Eq. (14) can be solved for the velocity,
( )( )( ) ( )( )
( )( ), ,
,
t t t t tt
t tψ ψ
ψ
× ∇ − ∂ ∂=
∇
E r B rv
B r . , (15)
where
( )( ) ( )( ),t t t tψ ≡ E r .B r , . (16)
In principle, one can compute the trajectory from an initial position ( )0r by integrating (15). Some care is
required in its application however. The fields and B can be regarded as given continuous functions of
space and time, (even in the direct-action theory), for as long as the charge in question is not moving
superluminally. In that case, though the charge is at all times generating new fields of its own, these fields
are never intersected by the world line of the charge in question. By virtue of the action (12), this remains true
even as . In this case of sub-luminal motion the fields E and B in (15) are those of other charges,
which, in a first iteration, can possibly be regarded as independent of the charge in question.
E
0∆ →
At superluminal speeds however, the particle’s own fields become visible to itself (emanating from
future and past positions). Correspondingly, the fields in (15) will then contain a contribution from the
particle’s own path, including the fields from a point infinitesimally nearby, as shown in Figure 3. This self-
action generates infinite self-fields. They will appear on the right had side of (15) and can be analyzed, using
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(7), in terms of the particle’s own acceleration, evaluated as ,r 0∆ → . The self-action will result in Eq. (15)
acquiring an additional term which will be interpretable as mass times acceleration, where the mass is
proportional to 1/ [9]. ∆
Classical Zitterbewegung
Several authors, including Barut and Zanghi [19], Hestenes [20-22], Rodrigues, Vaz, Recami, and Salesi
[23,24] have championed a literal interpretation of zitterbewegung of the Dirac electron. The Dirac ‘charge
point’ moves at light speed, though, as Hestenes points out, its center of mass does not. If interpreted
classically, one infers that the charge point must be massless, in accord with the presentation above, and the
favored action (12). If so, the observed mass must come from interactions with other charges. Taking into
account that the mean free path of a photon is of the order of the Hubble radius, there would appear to be no
opportunity for emergence of a universal common value for the electron mass from mutual electromagnetic
interactions. That conclusion is predicated on the correctness of the Maxwell theory, however. In the direct
action version of EM there is an opportunity for an equilibrium background field maintained through time-
symmetric fields. Supposing these fields exist and maintain the electron mass, then at 0K, they should be
consistent with purely zitterbewegung motion.
Equation (15) does not obviously demand solutions have speed c, though such a possibility cannot be
ruled out once time-symmetric far-field expressions for the fields are inserted. If zitterbewegung motion is
demanded and (15) does not deliver (very near) light speed motion for tardyons, one might consider
augmenting the action (12) with a small mechanical mass:
( ) ( )( ) ( ) ( ) ( )( )( ) ( )22 2 2
,
11 1
2 j k j k j k bare jj k j
I e e dt dt t t t t t t m dtδ′ ′ ′ ′= − − − − − + ∆ − −∑ ∑∫ ∫ ∫v .v x x v t . (17)
and let . This would restore the Newton-Lorentz equation to its familiar form: 0barem →
( )( ) ( )( ) ( ) ( )(( ,bare
d tm e t t t
dtγ
= + ×v
E r v B r )),t t . (18)
As one would expect 0barem → ( ) 1t →v as required, though this would have to be proved. A problem with
(17) however is that it destroys the promising superluminal / quark-like behavior; speeds ( ) 1t >v cause the
action to become imaginary. In the event that (15) does not deliver near light speed motion, other possibilities,
such as ‘softening’ the delta function, using for example, the Lorentzian
( ) ( )2 20
lim xx
δπΓ→
Γ=
+ Γ, (19)
may have to be considered.
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4. Summary We have presented some arguments in favor of the time-symmetric direct-action version of classical
electromagnetism. A novel property of the model presented here is the absence of mechanical mass in the total
action. Charged particles must instead acquire their inertial mass from electromagnetic interactions only. In
the case of sub-luminal and near-light-speed particles, an action has been presented wherein self-action is
absent, and therefore the bare electromagnetic mass is zero. For such particles inertial mass must arise from
electromagnetic interaction with distant charges, meditated in this case by direct-action fields. In the case of
super-luminal and time-reversing particles, the same action generates infinite self-action, and then the bare
electromagnetic mass becomes infinite. These particles can continue to effectively interact however, due to the
presence of infinite forces on the Cerenkov cone. The possibility was discussed that these two classes of
classical particles may correspond to leptons and quarks.
References
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10. R. P. Feynman, Phys. Rev. 76 (1949) 749.
11. M. Ibison, Fizika A 12 (2004) 55.
12. M. Ibison, in: A. Chubykalo et al (Eds.) Has the Last Word Been Said on Classical Electrodynamics?, Rinton Press, 2004.
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17. J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, Inc., 1998.
18. R. Mignani and E. Recami, Nuovo Cimento 30A (1975) 533.
19. A. O. Barut and N. Zanghi, Phys. Rev. Lett. 52 (1984) 2009.
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Table 1 Title: Solutions of the equation 2 21 tan 1 1− −v v = −
mode number v
1 2.972
2 6.202
3 9.371
4 12.526
5 15.676
large n nπ
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Figure Captions Figure 1
Regularized self-action. Ordinarily, the sub-luminal and superluminal charges self-interact by self-intersecting their own light cones at their present location. This can be treated analytically by replacing the light cone with hyperboloids, whereupon the self-action becomes finite. Later, one lets and the hyperboloids becomes light cones once again.
0∆ →
Figure 2
Self-consistent motion of two superluminal charges in orbit about a common origin. The orbits are quantized due to the requirement that EM interaction also take place on the Cerenkov cone. Shown here are the first two ‘modes’.
Figure 3
Regularized self-action for superluminal charges, and zero self-action for sub-luminal charges. The latter have no electromagnetic mass. As , the hyperbola becomes the light cone. 0∆ →